10 SCREEN 12
20 DEFDBL A-Z
50 PRINT "                Integration by Parts"
60 PRINT "The formula starts with the Product Rule for derivatives"
70 PRINT "=(f(x)g(x))'=f(x)g'(x)+f'(x)g(x)"
80 PRINT "                                  "; CHR$(244); "            "; CHR$(244)
90 PRINT "Integrate both sides for f(x)g(x)="; CHR$(245); "f(x)g'(x)dx+"; CHR$(245); "f'(x)g(x)dx"
100 PRINT "          "; CHR$(244)
110 PRINT "Solve for "; CHR$(245); "f(x)g'(x)dx getting"
120 PRINT CHR$(244); "                     "; CHR$(244)
130 PRINT CHR$(245); "f(x)g'(x)dx=f(x)g(x)-"; CHR$(245); "f'(x)g(x)dx"
140 PRINT "The integration by Parts formula is often expressed as follows:"
150 PRINT "Let u=f(x) du=f'(x)dx dv=g'(x)dx v=g(x)"
160 PRINT "                        "; CHR$(244); "       "; CHR$(244)
170 PRINT "Then the formula becomes"; CHR$(245); "udv=uv-"; CHR$(245); "vdu."
180 PRINT "To integrate by parts, strategically choose u, dv and then apply the formula."
190 PRINT "                  "; CHR$(244)
200 PRINT "Example: evaluate "; CHR$(245); "xcos(x)dx"
210 PRINT "Let u=x du=dx dv=cos(x) v=sin(x)"
220 PRINT CHR$(244); "          "; CHR$(244); "       "; CHR$(244); "            "; CHR$(244)
230 PRINT CHR$(245); "xcos(x)dx="; CHR$(245); "udv=uv-"; CHR$(245); "vdu=xsin(x)-"; CHR$(245); "sin(x)dx=xsin(x)+cos(x)+C"
235 PRINT "                    "; CHR$(244)
240 PRINT "Example 2: evaluate "; CHR$(245); "arctan(x)dx Let u=arctan(x) du=1/1+x^2dx dv=dx v=x"
245 PRINT CHR$(244); "                       "; CHR$(244)
250 PRINT CHR$(245); "arctan(x)dx=xarctan(x)-"; CHR$(245); "x/1+x^2dx=xarctan(x)-(1/2)log(1+x^2)"
260 SYSTEM